and where \(r\)  is the exponential growth rate of the expected incidence. A straight forward way of estimating \(F_R(G)\)  (and hence \(R_0\) ) would thus be to replace \(\varphi _G\)  by \(\hat{\varphi }_{G,n}\)  in (REF ), where \(\hat{\varphi }_{G,n}\)  is a non-parametric estimator of \(\varphi _G\) . This strategy is often called plug-in estimate and is of course not limited to \(F_R\)  but it can be applied to any function \(F\)  of the density \(\varphi _G\) , for instance mean value or quantiles. In the following we will call such functions \(F\)  features of \(\varphi _G\). In practice it is not always (or rarely) possible to make observations of the generation time directly because the infection times cannot be observed (cf. [1]}). However, the urgency of the situation requires estimation with imperfect data ([2]}). Sometimes there is additional information about the infection time available (e.g. if a person was only during a certain window \([0,W]\)  in a region where the virus is circulating, see [3]}, [4]} for examples related to travel to and from Wuhan in the early days of the pandemic). This leads to observations of the infection times with measurement error from which we have to infer \(\varphi _G\) . Other methods for estimation of \(R_0\)  make sometimes implicit assumptions about the generation time (cf. [5]}). Testing for such assumptions can be realized within the same framework if the test statistic can be written as a function of an estimate \(\hat{\varphi }_{G,n}\) , e.g., similarly to [6]}: In this case the feature \(F\)  of \(\varphi _G\)  of interest would be the test-statistic. In this paper we aim to do the first step to tackle this type of problems: Consistent, non-parametric estimation of the generation time density \(\varphi _G\) . Based on such an estimator the theory for semi-parametric inference methods about features of \(\varphi _G\)  like \(R_0\)  or test statistics can be developed. We mentioned above that this is a measurement error problem. In order to illustrate that the issue cannot be simply resolved, we need to be more specific about the involved quantities.
